Realized skewness at high frequency and link to
conditional market premium
Zhi Liub,c , Kent Wanga , Junwei Liua
a
The Wangyanan Institute for Studies in Economics, Xiamen University, China
b
Department of Mathematics, University of Macau, Macau, China
c
Corresponding author: Zhi LIU, Email: [email protected]
Abstract
We propose a reliable new estimator for realized skewness which is robust to microstructure noise at ultra-high frequency level. Asymptotic theory for the new
estimator has been de- rived. Simulation example verifies its superior performance.
We apply the new estimator to tick data of the S&P 500 index for forecasting equity premium in the U.S. market from 1990-2011 and find that it has significant
forecast-ability both in-sample and out-of-sample. We also show that the new skewness measure plus the variance risk premium provides right decomposition for the
skewness risk. We thus provide evidence that realized skewness links to conditional
market premium.
Keywords: , High-frequency, Jump, Noise, Skewness, Stock return prediction
1. Introduction
Traditional approaches of asset pricing concentrate mainly on the first and second moment of returns. But there is considerable evidence that the third moment
may also be important. Barro (2009) argues that the inclusion of the possibility
of rare disasters (extreme negative skewness) in an otherwise reasonably standard
representative-agent economy, generates equity premia and risk free rates that accord with observation. The empirical literature for skewness going back to Kraus
and Litzenberger (1976), and including more recently Das and Sundaram (1997),
Harvey and Siddique (2000), Ang et al. (2006), Chen et al. (2006) and Xing et al.
(2010), suggests that the asymmetry of the returns distribution both for individual stocks and for the equity market as a whole is important for asset pricing and
investment management.
Different skewness measures have been proposed in the literature but it is generally admitted that skewness is hard to measure precisely. Major existing skewness
proxies with theoretical foundations include risk neutral skewness measure advocated by Bakshi et al., realized skewness measures used in such as Amaya et al.
(2011)(ACJV (2011) hereafter)1 , Neuberger (2012) and skewness premium by combining implied and realized skewness. The difficulty for measuring skewness precisely
1
Note that the skewness measure in ACJV (2011) is standard skewness measure according to
skewness definition but uses high frequency data. In the current study, ACJV (2011) is one example
we use among many who apply such measure in empirical works.
Preprint submitted to 2013 World Statistics Congress
April 24, 2013
lies in the fact that people know little about the determinants or sources of skewness
and the understanding of the issue comes from limited empirical evidence which are
mainly based on hypothesis or even speculations2 .
The recent advent of readily-available high-frequency data, however, has spurred
a renewed interest into alternative ways for more accurately estimating skewness.
It is generally found that measures based on high frequency data can improve the
estimation and result in far better empirical performances (see, e.g. ACJV 2011). It
is believed that one needs to be able to exploit high frequency data to improve the
estimation of skewness if one seeks to understand the dynamics of skewness of period
returns Neuberger (2012). However, the realized skewness measures proposed based
on high frequency data are mostly just proxies with limited validity due to lack
of theoretical foundation and empirical considerations. For example, the realized
skewness measure appeared in ACJV (2011), which is in terms of the ratio of the
third cumulant and the 1.5th power of the second cumulant, has not considered any
of the common factors in the high frequency estimation such as microstructure noise
and jumps etc. They neither provide any theoretical proof for the unbiasedness
and consistency of the proposed empirical estimator.Neuberger (2012) provides a
very interesting ex-post estimator for skewness and we believe it is by far the most
theoretically justified measure for realized skewness. But Neuberger (2012) only
provides unbiasedness proof for the estimator without the consistency validation. In
addition, due to model setting, his measure does not accommodate important factors
such as noise in the high frequency data and hence can not be applied to the tick
data if needed. This leads us to believe that most of the current empirical measures
for ex-post skewness based on high frequency data are in fact rough indicators for the
“true” skewness in high frequency. There is great need to improve those estimators
based on theoretical and empirical considerations, which is what we propose to do
in the current study.
Based on the existing realized skewness measure, we derive the asymptotic properties for it and then develop our own measure of realized skewness accounting for
microstructure noise effect in the data. Simulation study is also performed to investigate finite sample properties of the new measure and it is found that the proposed
new estimator can produce far more accurate estimation for the realized third moment that is quite close to the ideal situation. We then apply the new method to
the tick data of the S&P 500 index from 1990-2011 in forecasting near future excess
equity market returns. We find that the new estimator for realized skewness based
on tick data significantly forecasts one-month-ahead excess equity market returns
and such result is obtained controlling for commonly used stock return predictors
in the literature including the powerful variance risk premium (VRP). The forecast
ability of our skewness measure remains significant at 5% in the out-of-sample prediction and shows considerable economic importance. This sheds new light on the
recognized puzzle that skewness measures only have cross-sectional pricing effects
while little time series performance has been documented before. The result thus
suggests that realized skewness is an effective predictor for stock market returns and
2
However, it is not too difficult to understand at intuitive level that asset return skewness is
caused by the jumps in the returns along with the leverage effect. Such point of view can be found
from works such as Das and Sundaram (1997) and Neuberger (2012).
controlling for noise can realistically reveal such effectiveness. In a further attempt
to explore the sources of documented predictability of the realized skewness in the
current study, we find the new realized skewness measure subsumes to a certain degree the market momentum effect in the short run. The finding that both our new
realized skewness measure and the variance risk premium are significant in predicting aggregate stock market premium indicates that the new measure provides an
independent channel for stock return prediction and we supply the literature with
a clean decomposition of the skewness risk into jump and leverage effect components. We therefore shed new light on both the time series pricing performance of
the skewness and the hypothesis regarding the sources of the skewness risk.
We organize the rest of the paper as following. Section 2 provides theoretical
discussions for deriving asymptotic properties of the existing high frequency based
estimator for skewness and kurtosis as in ACJV (2011) as well as theories for developing our new skewness measure that accounts for microstructure noise; Section 3
provides discussions for the simulation studies; Section 4 presents empirical illustrations for applying the new skewness measures in forecasting excess S&P 500 index
returns; Section 5 provides further discussion for the results obtained and Section 6
concludes. We postpone all the necessary theoretical proof and justification for our
methodology to the Appendix.
2. Methodology
2.1. The Model and Estimators
We define two processes {Xt , t ≥ 0} and {Yt , t ≥ 0}, both are adapted to the
filtered probability space (Ω, F, Ft , P ) as follows:
Z
t
Z
bs ds +
Xt =
0
t
Z
σs dWs , Yt =
0
t
Z
bs ds +
0
t
σs dWs +
0
X
∆s Y,
(1)
s≤t
where {bt , 0 ≤ t ≤ T } is adapted locally bounded processes, {σt , 0 ≤ t ≤ T } is an
deterministic càdlàg volatility process. ∆s Y = Ys − Ys− is the jump of Y at time
s, if there is no jumps, it is simple zero. We assume that the jump of Y arrives via
a finite activity jump process, which includes commonly used compound Poisson
process as a special case. We observe the processes at some discrete time points ti ,
0 = t0 ≤ ti ≤ tn = T , that is, we have the increments as ∆Zi = Zti+1 − Zti for either
Z = X or Z = Y . Hence, in the following context, we use Z referring to either as
X or Y , namely, when Z = X, it has continuous path, Z = Y means there is jumps
in the path.
2.2. Microstructure-noise free estimator
In this section, we investigate the properties of the estimator proposed by ACJV
(2011) under both X and Y . We assume that the log price follows the process Z,
and then the increments correspond to the log returns for each firm. ACJV (2011)
proposed the realized daily skewness and realized kurtosis as the following:
P
√ Pn
n ni=1 (∆Zi )4
n i=1 (∆Zi )3
RDSkew =
,
RDKurt
=
,
(2)
3/2
RDV arT2
RDV arT
where RDV art defines well-known realized volatility RDV arT =
now study the asymptotic properties of the estimators.
Pn
2
i=1 (∆Zi ) .
We
Theorem 1. Z defines a Itô process, and ∆Zi = Zti+1 − Zti . Then
1. When Z = X, we have
n
X
i=1
2
Z
T
(∆Zi ) →P
σt2 dt, n
0
Z
n
n
X
X
3
4
(∆Zi ) →D N, n
(∆Zi ) →P 3
i=1
i=1
T
σt4 dt. (3)
0
Where, N is a random variable with normal distribution, and →D defines the
convergence in distribution and →P is the convergence in probability.
2. When Z = Y , we have
Z
n
X
(∆Zi )2 →P
T
σt2 dt +
0
i=1
X
(∆t Y )2 ,
t≤T
n
n
X
√ X
3
n
(∆Zi ) →P ∞, n
(∆Zi )4 →P ∞.
i=1
(4)
(5)
i=1
Remark 2. From the above results, we have that asymptotically, both of RDSkew
and RDKurt are not well-defined in the jump-diffusion model (tend to infinity asymptotically). In the diffusion model, the RDSkew is not reasonable since
RDSkew →P 0. Considering of jump is accepted both theoretically and empirically, instead of pursuing more sophistical estimator for the diffusion model, we aim to
propose some reasonable estimators for jump-diffusion model.
To deal with this problem, we propose to use the following new estimators for
skewness and kurtosis:
Pn
Pn
4
3
i=1 (∆Zi )
i=1 (∆Zi )
,
J
−
RDKurt
:=
.
(6)
J − RDSkew :=
3/2
RDV art2
RDV art
Here we use J to emphasize the “jumps”. The following theorem specifically describes the asymptotic performance of the estimators proposed above.
Theorem 3. Z = Y is defined as in (1), we have
P
3
t≤T (∆t Z)
J − RDSkew →P R T
,
P
( 0 σt2 dt + t≤T (∆t Z)2 )3/2
P
4
t≤T (∆t Z)
J − RDKurt →P R T
P
( 0 σt2 dt + t≤T (∆t Z)2 )2
(7)
(8)
We set the limits as the measure of skewness and kurtosis under the high frequency situation with the jump-diffusion driven underlying. We aim to estimate this
quantity (note that both of them are random variables) using noisy high frequency
data.
2.3. Estimation with microstructure noise
The above proposed estimators can not typically be used to the financial data
with higher frequency, say tick-by-tick, or even one-minute, because high frequency
data inevitably is contaminated by microstructure noise, which is usually induced by
bid-ask bounce, rounding error, etc. More specifically, we observe the noisy process
Mi = Zi + i , rather than the latent process Z, at points ti , 0 ≤ i ≤ n.
To deal with the noisy data, existing approaches including two-time scales method
proposed by Zhang et al. (2005), preaveraging method studied by Jacod et al. (2009)
and further developed by Jacod et al. (2010), and realized kernel method suggested
by Barndorff-Nielsen et al. (2008). In this paper, we use the preaveraging approach
as described in Jacod et al. (2010). Here we briefly describe the procedure, although
it is well-known already. Suppose that the latent process or the log price of a security is Z, but we only can observe M = Z + , where ⊥ Z. As shown in Zhang et al.
(2005), if the noise left untreated, will plays dominant role in the estimator. To
reduce the effect of microstructure noise, a block with length kn is constructed. For
example, in ith block, we use ∆Mi , ∆Mi+1 , · · · , ∆Mi+kn −1 to construct a sequence
of preaveraging returns for 1 ≤ i ≤ n − kn :
∆ni,kn M (g) =
kn
X
j=1
k
g(
n 2
X
j
j
j−1
(g( ) − g(
)∆Mi+j , ∆ni,kn M̄ (g) =
))∆Mi+j ,
kn
kn
kn
j=1
with a non-negative piece-wise
R 1 differentiable Lipschitz function g, satisfying g(x) = 0
when x 6∈ (0, 1) and ḡ(p) = 0 g p (x)dx > 0. Consequently, the effect of microstructure noise is significantly reduced to a matched order of the latent increment.
Our estimator is constructed basing on preaveraging returns. We collect all the
preaveraging returns by the following way:
Un (M, p) =
n−k
Xn
(∆ni,kn M (g))p , Ūn (M ) =
i=1
n−k
Xn
∆ni,kn M̄ (g)
(10)
i=1
We have the following asymptotics for the above statistic.
Theorem 4. Z = Y is defined as in (1), we have
X
1
Un (M, p) →P ḡ(p)
(∆s Y )p For all p > 2,
kn
(11)
s≤T
Z T
X
1
1
Un (M, 2) −
Ūn (M ) →P ḡ(2)[Y, Y ]T = ḡ(2)(
σs2 ds +
(∆s Y )2 ).(12)
kn
2kn
0
s≤T
Inspired by above results, using the Slustky’s theorem, we obtain a consistent estimator of skewness and kurtosis which we set in the earlier section.
Theorem 5. Z = Y is defined as in (1), we have
P
√
3
kn Un (M, 3)
(ḡ(2))3/2
s≤T (∆s Y )
→
, (13)
R
P
P
T
ḡ(3) (Un (M, 2) − Ūn (M )/2)3/2
( 0 σs2 ds + s≤T (∆s Y )2 )3/2
P
4
(ḡ(2))2
kn Un (M, 4)
s≤T (∆s Y )
→
.
(14)
P RT
P
ḡ(4) (Un (M, 2) − Ūn (M )/2)2
( σs2 ds +
(∆s Y )2 )2
0
s≤T
(9)
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